Degasperis-Procesi方程的孤立尖波解

利用动力系统的定性分析理论对D egasperis-P rocesi方程的孤立尖波解进行了研究.给出了D e-gasperis-P rocesi方程对应行波系统的相图分支,利用相图获得了孤立尖波解和周期尖波解的解析表达式,通过数值模拟给出了部分解的图像.     

带色散项的Degasperis-Procesi方程的孤立尖波解

用动力系统的定性分析理论研究了带有色散项的Degasperis-Procesi方程的孤立尖波解.在一定的参数条件下,利用Degasperis-Procesi方程对应行波系统的相图分支从两种不同方式给出了孤立尖波解的表达式.     

小展弦比波的Green-Naghdi渐进模型的行波解

本文研究了小展弦比波的Green-Naghdi渐进模型. 利用平面自治系统的稳定性分析方法, 在不同的参数条件下, 讨论了它的行波系统的分岔并且给出了对应的相图, 得到了光滑周期波解, 广义扭波解, 广义反扭波解, 广义紧波解, 周期尖波解, 孤波解和孤立尖波解的精确表达式. 进一步, 通过数学软件Maple模拟了这些解.     

浅水中度振幅孤立波解的分支

利用平面动力系统分支方法研究浅水中度振幅方程的定性行为和孤立波解.给出了系统在不同参数条件下的相图.获得了光滑孤立波、cuspon解和周期波解的隐式表达式.对方程的光滑孤立波解、cuspon解和周期波解进行了数值模拟.获得的结果完善了相关文献已有的结果.     

小展弦比波的Green-Naghdi渐进模型的行波解（英文）

本文研究了小展弦比波的Green-Naghdi渐进模型.利用平面自治系统的稳定性分析方法,在不同的参数条件下,讨论了它的行波系统的分岔并且给出了对应的相图,得到了光滑周期波解,广义扭波解,广义反扭波解,广义紧波解,周期尖波解,孤波解和孤立尖波解的精确表达式.进一步,通过数学软件Maple模拟了这些解.     

立方非线性微结构固体中的对称孤立波及存在条件

考虑固体材料的宏观尺度立方非线性效应、微尺度立方非线性效应以及微尺度频散效应并根据修正的Mindlin理论,建立了一维微结构固体中纵波传播的一种新模型.用动力系统的定性分析方法,证明了适当条件下立方非线性微结构固体中可存在对称钟型孤立波和反钟型孤立波,并给出了两种孤立波的存在条件.用数值方法分析了微尺度立方非线性效应对钟型与反钟型孤立波的影响,结果显示随着微尺度非线性效应的增强(或负增强),两种孤立波的宽度变窄(或变宽)而幅度保持不变.     

非线性波方程的精确孤立波解

立了一种求解非线性波方程精确孤立波解的双曲函数方法,并在计算机代数系统上加以实现,推导出了一大批非线性波方程的精确孤立波解.方法的基本原理是利用非线性波方程孤立波解的局部性特点,将方程的孤立波解表示为双曲函数的多项式,从而将非线性波方程的求解问题转化为非线性代数方程组的求解问题.利用吴消元法或Gröbner基方法在计算机代数系统上求解非线性代数方程组, 最终获得非线性波方程的精确孤立波解,其中有很多新的精确孤立波解.     

复杂固体并式微结构模型及孤立波的存在性

把复杂固体看作具有两种不同性质的微结构,进而考虑两种微尺度非线性效应,建立了描述复杂固体运动的并式微结构非线性模型.利用动力系统的定性分析理论和分岔理论,证明了在一定条件下并式微结构固体中可以存在一类非对称孤立波并给出了其存在条件.分析表明两种微尺度非线性效应同时影响孤立波的对称特性,微尺度非线性效应越强,孤立波的非对称特性越明显.最后用数值方法进一步验证了定性分析结果.     

广义Gilson-Pickering方程的分支解

本文利用动力系统方法和奇行波方程理论研究广义Gilson-Pickering方程的动力学行为和行波解.利用软件画出了给定参数条件下系统的相图分支,得到了孤立波解、扭结波解和反扭结波解、不可数无穷多破缺波解、光滑周期波解和非光滑周期尖波解、尖孤子解的存在性.在β≠1,p=2时,对于广义Gilson-Pickering方程不同的参数条件下,给出了保证上述解存在的条件及参数表示.     

非线性波方程准确孤立波解的符号计算

该文将机械化数学方法应用于偏微分方程领域，建立了构造一类非线性发展方程孤立波解的一种统一算法，并在计算机数学系统上加以实现，推导出了一批非线性发展方程的精确孤立波解．算法的基本原理是利用非线性发展方程孤立波解的局部性特点，将孤立波表示为双曲正切函数的多项式．从而将非线性发展方程（组）的求解问题转化为非线性代数方程组的求解问题．利用吴文俊消元法在计算机代数系统上求解非线性代数方程组，最终获得非线性发展方程（组）的准确孤立波解.     

Bifurcations and exact traveling wave solutions of the equivalent complex short-pulse equations

In this paper, we study the traveling wave solutions for a complex short-pulse equation of both focusing and defocusing types, which governs the propagation of ultrashort pulses in nonlinear optical fibers. It can be viewed as an analog of the nonlinear Schrodinger (NLS) equation in the ultrashort-pulse regime. The corresponding traveling wave systems of the equivalent complex short-pulse equations are two singular planar dynamical systems with four singular straight lines. By using the method of dynamical systems, bifurcation diagrams and explicit exact parametric representations of the solutions are given, including solitary wave solution, periodic wave solution, peakon solution, periodic peakon solution and compacton solution under different parameter conditions.     

Effects of Quadratic Singular Curves in Integrable Equations

In this paper, the effects of quadratic singular curves in integrable wave equations are studied by using the bifurcation theory of dynamical system. Some new singular solitary waves (pseudo‐cuspons) and periodic waves are found more weak than regular singular traveling waves such as peaked soliton (peakon), cusp soliton (cuspon), cusp periodic wave, etc. We show that while the first‐order derivatives of the new singular solitary wave and periodic waves exist, their second‐order derivatives are discontinuous at finite number of points for the solitary waves or at infinitely countable points for the periodic wave. Moreover, an intrinsic connection is constructed between the singular traveling waves and quadratic singular curves in the phase plane of traveling wave system. The new singular periodic waves, pseudo‐cuspons, and compactons emerge if corresponding periodic orbits or homoclinic orbits are tangent to a hyperbola, ellipse, and parabola. In particular, pseudo‐cuspon is proposed for the first time. Finally, we study the qualitative behavior of the new singular solitary wave and periodic wave solutions through theoretical analysis and numerical simulation.     

Singular periodic waves of an integrable equation from short capillary-gravity waves

The effects of parabola singular curves in the integrable nonlinear wave equation are studied by using the bifurcation theory of dynamical system. We find new singular periodic waves for a nonlinear wave equation from short capillary-gravity waves. The new periodic waves possess weaker singularity than the periodic peakon. It is shown that the second derivatives of the new singular periodic wave solutions do not exist in countable number of points but the first derivatives exist. We show that there exist close connection between the new singular periodic waves and parabola singular curve in phase plane of traveling wave system for the first time.     

Solitary wave families of a generalized microstructure PDE

Wave propagation in a generalized microstructure PDE, under the Mindlin relations, is considered. Limited analytic results exist for the occurrence of one family of solitary wave solutions of these equations. Since solitary wave solutions often play a central role in the long-time evolution of an initial disturbance, we consider such solutions here (via normal form approach) within the framework of reversible systems theory. Besides confirming the existence of the known family of solitary waves, we find a continuum of delocalized solitary waves (or homoclinics to small-amplitude periodic orbits). On isolated curves in the relevant parameter region, the delocalized waves reduce to genuine embedded solitons. The new family of solutions occur in regions of parameter space distinct from the known solitary wave solutions and are thus entirely new. Directions for future work are also mentioned.     

Bifurcations and exact travelling wave solutions of M-N-Wang equation

By using the method of dynamical systems to Mikhailov-Novikov-Wang Equation, through qualitative analysis, we obtain bifurcations of phase portraits of the traveling system of the derivative $\phi(\xi)$ of the wave function $\psi(\xi)$. Under different parameter conditions, for $\phi(\xi)$, exact explicit solitary wave solutions, periodic peakon and anti-peakon solutions are obtained. By integrating known $\phi(\xi)$, nine exact explicit traveling wave solutions of $\psi(\xi)$ are given.     

Bifurcations of travelling wave solutions for the generalized (2+1)-dimensional Boussinesq-Kadomtsev-Petviashvili equation

By using the bifurcation theory of dynamical systems, we study the generalized (2+1)-dimensional Boussinesq-Kadomtsev-Petviashvili equation, the existence of solitary wave solutions, compacton solutions, periodic cusp wave solutions and uncountably infinite many smooth periodic wave solutions are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of the above waves are determined.     

Bifurcations of travelling wave solutions for a class of nonlinear fourth order variant of a generalized Camassa–Holm equation

In this paper, by using the bifurcation theory of dynamical systems for a class of nonlinear fourth order variant of a generalized Camassa–Holm equation, the existence of solitary wave solutions, breaking bounded wave solutions, compacton solutions and non-smooth periodic wave solutions are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of the above waves are determined.     

Multi-compacton and double symmetric peakon for generalized Ostrovsky equation

In this paper, the generalized Ostrovsky equation is introduced. Using a direct and effective method, some new solitary solutions to the generalized Ostrovsky equation, such as compacton solutions, multi-compacton solutions and compact-like kink solutions can be obtained. The homogenous balance (HB) method is used to obtain the Backlund transformation. And some new solitary solutions, particularly new double symmetric peakon solutions, are given by the transformation.     

Smooth and non-smooth traveling wave solutions of some generalized Camassa–Holm equations

In this paper we employ two recent analytical approaches to investigate the possible classes of traveling wave solutions of some members of a recently-derived integrable family of generalized Camassa–Holm (GCH) equations. A recent, novel application of phase-plane analysis is employed to analyze the singular traveling wave equations of three of the GCH NLPDEs, i.e. the possible non-smooth peakon and cuspon solutions. One of the considered GCH equations supports both solitary (peakon) and periodic (cuspon) cusp waves in different parameter regimes. The second equation does not support singular traveling waves and the last one supports four-segmented, non-smooth M-wave solutions.Moreover, smooth traveling waves of the three GCH equations are considered. Here, we use a recent technique to derive convergent multi-infinite series solutions for the homoclinic orbits of their traveling-wave equations, corresponding to pulse (kink or shock) solutions respectively of the original PDEs. We perform many numerical tests in different parameter regime to pinpoint real saddle equilibrium points of the corresponding GCH equations, as well as ensure simultaneous convergence and continuity of the multi-infinite series solutions for the homoclinic orbits anchored by these saddle points. Unlike the majority of unaccelerated convergent series, high accuracy is attained with relatively few terms. We also show the traveling wave nature of these pulse and front solutions to the GCH NLPDEs.     

Solitary Waves, periodic peakons, pseudo-peakons and compactons given by three ion-acoustic wave models in electron plasmas

The nonlinear ion-acoustic oscillations models are governed by three partial differential equation systems. Their travelling wave equations are three first class singular traveling wave systems depending on different parameter groups, respectively. By using the method of dynamical system and the theory of singular traveling wave systems, in this paper, it is shown that there exist parameter groups such that these singular systems have solitary wave solutions, pseudo-peakons, periodic peakons and compactons as well as kink and anti-kink wave solutions. The results of this paper complete the studies of three papers [5,13] and [14].